The population of Cirque City was 46,229 in 1999 and was 55,680 in 2010. If the population was growing exponentially, what will the population be in2020?

Start by subtracting 46,229 from 55,680 = 9,451 total growth.

Presuming the growth is a constant of the average: divide 9,451 / 11 = 859.18 growth per year. (2010 - 1999 = 11 years)

859.18 x 10 = 8,591 total growth ---- 2020 - 2010 = 10 years

Population in 2010 = 55,680 + 8,591 = 64,271 total population in 2020

Exponentially just means quickly.

Answer Link

MrRoyal MrRoyal · Answer 2 · 2021-08-10T14:59:08+02:00

When a function is defined by an exponential growth, it means the rate of the function is greater than 1. The population of the city in 2020 is 65865.

An exponential function is represented as:

[tex]y = ab^x[/tex]

In this case:

[tex]y \to[/tex] population

[tex]x \to[/tex] years since 1999

In 1999:

[tex]x = 1999 - 1999[/tex]

[tex]x= 0[/tex]

So, we have:

[tex](x_1,y_1) = (0,46229)[/tex]

In 2010

[tex]x = 2010 - 1999[/tex]

[tex]x = 11[/tex]

So, we have:

[tex](x_2,y_2) = (11,55680)[/tex]

Using [tex]y = ab^x[/tex]

[tex](x_1,y_1) = (0,46229)[/tex] is represented as:

[tex]46229 = ab^0[/tex]

[tex]46229 = a*1[/tex]

[tex]a =46229[/tex]

[tex](x_2,y_2) = (11,55680)[/tex] is represented as:;

[tex]55680 = ab^{11}[/tex]

Substitute [tex]a =46229[/tex]

[tex]55680 = 46229 * b^{11}[/tex]

Divide both sides by 46229

[tex]1.204 = b^{11}[/tex]

Take 11th roots of both sides

[tex]1.017= b[/tex]

[tex]b =1.017[/tex]

So, we have:

[tex]a =46229[/tex]

[tex]b =1.017[/tex]

The function becomes:

[tex]y = ab^x[/tex]

[tex]y = 46229 * 1.017^x[/tex]

The value of x in 2020 is:

[tex]x = 2020 - 1999[/tex]

[tex]x = 21[/tex]

So; the population in 2020 is:

[tex]y = 46229 * 1.017^{21}[/tex]

[tex]y = 65865[/tex]

Hence, the population of the city in 2020 is 65865.